Evaluate each expression for a = 2, b = 3 and Express values in rational form.

Step 1: substitute a = 2 and b = 3 into the expression

\(\sqrt{\frac{9\left(3\right)^3\left(2\cdot 3\right)^2}{\left(2^2\cdot 3^3\right)^3}}\)

Step 2: distribute the exponents within the parentheses where applicable

\(\sqrt{\frac{9\left(3\right)^3\cdot 2^2\cdot 3^2}{\left(2^2\right)^3\cdot \left(3^3\right)^3}}\)

Step 3: simplify the numerator within the radicand by combining the factors containing base 3. Also, use the power to a power exponent law for the denominator

\(\sqrt{\frac{9\left(3\right)^5\cdot 2^2}{2^6\cdot 3^9}}\)

Step 4: simplify the radicand further by cancelling out factors of 2 and 3

\(\sqrt{\frac{9}{2^4\cdot 3^4}}\)

Step 5: distribute the radical

\(\frac{\sqrt{9}}{\sqrt{2^4\cdot 3^4}}\rightarrow \ \frac{3}{\sqrt{2^4}\cdot \sqrt{3^{4\ }}}\rightarrow \ \frac{3}{4\cdot 9}\rightarrow \ \frac{3}{36}=\frac{1}{12}\)